I agree fully with Kathie's main point. But just to disagree with a detail or two:
1. The big question is: how many generations back do you have Nicketti Powhatan in your tree? Her birth year is only a proxy for generational difference, but, for instance, I have a 4th-great grandfather who was born in 1746. There are a couple of men who are reportedly still living as of 2019 who are grandchildren of a man born in 1790 (that man being US President John Tyler). So you can get pretty far back without going through too many generations if you are lucky.
2. Let's say we are talking about 9 generations back -- close to the 10 Kathie mentioned -- so we're talking about 7th great-grandparents. The amount of DNA shared with a 7th great-grandparent is the same as that expected between 4th cousins. That is, you would share typically about 13cM, so a detectable amount ... if you could test your 7th great grandparent, but unless you have a paleogenetics lab of your own, this is all hypothetical. You'll share more than this with some 7th-great-grandparents and less than others, but it's estimated also that you'd have a little over 50% chance of sharing detectable DNA with a typical 7th-great grandparent. So saying "infinitesimal", even at one generation further back, is an exaggeration.
Given another 9-generation descendant of that 7th great-grandparent, there is then about a 25% chance that both you and that cousin share detectable DNA with that ancestor. The problem is that it is then exceedingly unlikely that you share the same piece of DNA. It may be, for instance, that you share the first part of chromosome 7 with your 7th-great grandparent while this other cousin just shares a little chunk about 2/3 the way along chromosome 18. So you didn't inherit the same piece of DNA, and usually when that happens, you have no other shared DNA either -- you won't be matches.
Fortunately, if you share DNA with an 7th great-grandparent, typically you will have thousands of living cousins who descend from that same person. If you share a 13cM segment with the 7th great-grandparent, then because you can fit only a few hundred 13cM segments into your wholegenome, the chances aren't terrible that one of those cousins will also have inherited a detectable segment that significantly overlaps your 13cM segment.
But will that cousin have tested? Probably not. If they have, will they have published their tree? Probably not. If they have published their tree, will it extend back and include that 8th great-grandparent? Probably not.
But you can repeat this same game with your 1024 eigth great grandparents, and there is probably a non-infinitesimal chance that you can confirm one 8th cousin with a segment inherited by descent. But finding that person is like finding a needle in a haystack. I've found lots of individual matches to 8th cousins and beyond, but none has yet triangulated and I assume that they are all false matches, or IBP, or at least may have come down a line other than the one I think -- maybe with a more recent common ancestor.
Back to Jennifer: you want to use autosomal DNA to confirm descent from a *specific* ancestor rather than the much weaker goal of "some one of my many ancestors back 9 or so generations." That is exceedingly unlikely. Maybe if a thousand people played this same game, one could get lucky. But chances are, you are not that person (and neither am I).
For me -- I have confirmed lots of 5th cousins through individual matches, and most of those with good trees indicate there is no other reasonable place for the shared DNA to have come from than the ancestor we identified. Past this, it is still possible to confirm 6th cousins, maybe 7th, and maybe even 8th, but difficulty goes up exponentially with each generation back, and your probability of success goes down. But if you are one of the lucky people that can confirm an eighth cousin with a nice solid triangulation, know that you won't get to pick the ancestor you confirm in this manner. It will just be luck of the draw.